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Math Help - Application of strong operator convergence

  1. #1
    Junior Member
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    Application of strong operator convergence

    Hello!

    Any help with the following is really appreciated (hints, as well!):

    For f \in L^{1} ( \mathbb{R} ) and  s \in \mathbb{R} , let

     \hat{f} (s) = \int_{\mathbb{R}} f (t) e^{-ist} dt<br />

    Prove that  \hat{f} (s) \rightarrow 0<br />
as  |s| \rightarrow \infty<br />

    We're allowed to use the 'strong operator convergence' theorem for Banach spaces, as well as other standard results in Banach spaces.
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  2. #2
    A Plied Mathematician
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    That is the Riemann-Lebesgue Lemma. You can find the proof on page 103 of "Papa Rudin" (Real and Complex Analysis). You can also go here for an outline of the proof.
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